Optimal. Leaf size=297 \[ -\frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c)^3 \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}}+\frac {3 \sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c) \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}+\frac {(a d+b c)^2 e^{x (a d+b c)+a c+b d x^2}}{8 b^3 d^3}-\frac {x (a d+b c) e^{x (a d+b c)+a c+b d x^2}}{4 b^2 d^2}-\frac {e^{x (a d+b c)+a c+b d x^2}}{2 b^2 d^2}+\frac {x^2 e^{x (a d+b c)+a c+b d x^2}}{2 b d} \]
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Rubi [A] time = 0.64, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2244, 2241, 2240, 2234, 2204} \[ -\frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c)^3 \text {Erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}}+\frac {3 \sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} (a d+b c) \text {Erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}+\frac {(a d+b c)^2 e^{x (a d+b c)+a c+b d x^2}}{8 b^3 d^3}-\frac {x (a d+b c) e^{x (a d+b c)+a c+b d x^2}}{4 b^2 d^2}-\frac {e^{x (a d+b c)+a c+b d x^2}}{2 b^2 d^2}+\frac {x^2 e^{x (a d+b c)+a c+b d x^2}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2240
Rule 2241
Rule 2244
Rubi steps
\begin {align*} \int e^{(a+b x) (c+d x)} x^3 \, dx &=\int e^{a c+(b c+a d) x+b d x^2} x^3 \, dx\\ &=\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac {\int e^{a c+(b c+a d) x+b d x^2} x \, dx}{b d}-\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} x^2 \, dx}{2 b d}\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{4 b^2 d^2}+\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{2 b^2 d^2}+\frac {(b c+a d)^2 \int e^{a c+(b c+a d) x+b d x^2} x \, dx}{4 b^2 d^2}\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}-\frac {(b c+a d)^3 \int e^{a c+(b c+a d) x+b d x^2} \, dx}{8 b^3 d^3}+\frac {\left ((b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{4 b^2 d^2}+\frac {\left ((b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{2 b^2 d^2}\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {3 (b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}-\frac {\left ((b c+a d)^3 e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{8 b^3 d^3}\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b^2 d^2}+\frac {(b c+a d)^2 e^{a c+(b c+a d) x+b d x^2}}{8 b^3 d^3}-\frac {(b c+a d) e^{a c+(b c+a d) x+b d x^2} x}{4 b^2 d^2}+\frac {e^{a c+(b c+a d) x+b d x^2} x^2}{2 b d}+\frac {3 (b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{8 b^{5/2} d^{5/2}}-\frac {(b c+a d)^3 e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{16 b^{7/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 191, normalized size = 0.64 \[ \frac {e^{-\frac {(b c-a d)^2}{4 b d}} \left (2 \sqrt {b} \sqrt {d} e^{\frac {(a d+b (c+2 d x))^2}{4 b d}} \left (a^2 d^2-2 b d (-a c+a d x+2)+b^2 \left (c^2-2 c d x+4 d^2 x^2\right )\right )-\sqrt {\pi } \left (a^3 d^3+3 b^2 c d (a c-2)+3 a b d^2 (a c-2)+b^3 c^3\right ) \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )\right )}{16 b^{7/2} d^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 212, normalized size = 0.71 \[ \frac {\sqrt {\pi } {\left (b^{3} c^{3} + a^{3} d^{3} + 3 \, {\left (a^{2} b c - 2 \, a b\right )} d^{2} + 3 \, {\left (a b^{2} c^{2} - 2 \, b^{2} c\right )} d\right )} \sqrt {-b d} \operatorname {erf}\left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, b d}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )} + 2 \, {\left (4 \, b^{3} d^{3} x^{2} + b^{3} c^{2} d + a^{2} b d^{3} + 2 \, {\left (a b^{2} c - 2 \, b^{2}\right )} d^{2} - 2 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} e^{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )}}{16 \, b^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 250, normalized size = 0.84 \[ \frac {\frac {\sqrt {\pi } {\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} - 6 \, b^{2} c d - 6 \, a b d^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-b d} {\left (2 \, x + \frac {b c + a d}{b d}\right )}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{\sqrt {-b d}} + 2 \, {\left (b^{2} d^{2} {\left (2 \, x + \frac {b c + a d}{b d}\right )}^{2} - 3 \, b^{2} c d {\left (2 \, x + \frac {b c + a d}{b d}\right )} - 3 \, a b d^{2} {\left (2 \, x + \frac {b c + a d}{b d}\right )} + 3 \, b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} - 4 \, b d\right )} e^{\left (b d x^{2} + b c x + a d x + a c\right )}}{16 \, b^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 368, normalized size = 1.24 \[ \frac {x^{2} {\mathrm e}^{b d \,x^{2}+a c +\left (a d +b c \right ) x}}{2 b d}-\frac {\left (a d +b c \right ) \left (\frac {x \,{\mathrm e}^{b d \,x^{2}+a c +\left (a d +b c \right ) x}}{2 b d}+\frac {\sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{4 \sqrt {-b d}\, b d}-\frac {\left (a d +b c \right ) \left (\frac {\left (a d +b c \right ) \sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{4 \sqrt {-b d}\, b d}+\frac {{\mathrm e}^{b d \,x^{2}+a c +\left (a d +b c \right ) x}}{2 b d}\right )}{2 b d}\right )}{2 b d}-\frac {\frac {\left (a d +b c \right ) \sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{4 \sqrt {-b d}\, b d}+\frac {{\mathrm e}^{b d \,x^{2}+a c +\left (a d +b c \right ) x}}{2 b d}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.11, size = 267, normalized size = 0.90 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, b d x + b c + a d\right )} {\left (b c + a d\right )}^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}\right ) - 1\right )}}{\left (b d\right )^{\frac {7}{2}} \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}} - \frac {6 \, {\left (b c + a d\right )}^{2} b d e^{\left (\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}}{\left (b d\right )^{\frac {7}{2}}} + \frac {8 \, b^{2} d^{2} \Gamma \left (2, -\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, b d x + b c + a d\right )}^{3} {\left (b c + a d\right )} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}{\left (b d\right )^{\frac {7}{2}} \left (-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}\right )^{\frac {3}{2}}}\right )} e^{\left (a c - \frac {{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{16 \, \sqrt {b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 230, normalized size = 0.77 \[ \frac {{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}\,\left (\frac {a^2\,d^2}{8}-b\,\left (\frac {d}{2}-\frac {a\,c\,d}{4}\right )+\frac {b^2\,c^2}{8}\right )}{b^3\,d^3}+\frac {x^2\,{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}}{2\,b\,d}-\frac {x\,{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}\,\left (a\,d+b\,c\right )}{4\,b^2\,d^2}-\frac {\sqrt {\pi }\,{\mathrm {e}}^{\frac {a\,c}{2}-\frac {a^2\,d}{4\,b}-\frac {b\,c^2}{4\,d}}\,\mathrm {erfi}\left (\frac {\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x}{\sqrt {b\,d}}\right )\,\left (a^3\,d^3+3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-6\,a\,b\,d^2+b^3\,c^3-6\,b^2\,c\,d\right )}{16\,b^3\,d^3\,\sqrt {b\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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